How to Calculate the Number of Ways to Draw a Poker Hand

[Rayzen]

I am doing something wrong, when trying to calculate the number of hands that can be drawn, when dealing two pairs in a poker game, but can't see what it is, so I would appreciate being shown where my mistake is and--most importantly--WHY I am wrong, because I just can't see anything wrong with the way that I am doing it. My erroneous answer is exactly double what it should be, so I must be counting things twice.

Here is what I have (incorrectly) tried:

a. For the first pair, I choose one rank/denomination, then two suites: (13C1)(4C2)
b. Then, for the second pair, I again choose one rank/denomination from the remaining 12, then two suites: (12C1)(4C2)
c. Then, for the fifth card, I choose one rank/denomination from any of the remaining ranks/denominations, then one suite: (11C1)(4C1)
d. Finally, turning the crank, I get [(13C1)(4C2)][(12C1)(4C2)][(11C1)(4C1)] = (13)(6)(12)(6)(11)(4) = 247,104.

But the right way to do it is to choose both of the pairs' ranks/denominations simultaneously, like this:

[(13C2)(4C2)(4C2)][(11C1)(4C1)] = (78)(6)(6)(11)(4), which = 123,552.

Why do I have to choose both ranks/denominations at the same time, to start? What is wrong with my way of doing it?